Automorphic forms constructed from Whittaker vectors
type: Item
instance of: scholarly article
title: Automorphic forms constructed from Whittaker vectors
zbMATH Open document ID: 0692.10029
DOI: 10.1016/0022-1236(89)90059-1
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let G be a semisimple Lie group of split rank 1 and \(\Gamma\) a discrete subgroup of finite covolume. Forming a generalized Poincaré series of weak Whittaker functions attached to the nonunitary principal series a meromorphic family M(s) of functions on \(\Gamma\) \(\setminus G\) is obtained that satisfy all axioms of automorphic forms except the condition of moderate growth. It is shown that the principal part at a pole \(s_ 0\), Re \(s_ 0\geq 0\) is square integrable and that for a fixed K-type all automorphic forms up to a finite dimensional subspace are obtained. For the trivial K-type all automorphic forms are derived this way. The Fourier coefficients of the M-series are computed and a functional equation is derived. Trying to generalize this to higher rank leads to the observation that in certain cases (proved here for \(G=SO(n,1)^ d)\) the moderate growth condition for automorphic forms is redundant. This may be called a Koecher principle. The conjecture is stated that a Koecher principle holds for all G of rank\(\geq 2\) and \(\Gamma\) irreducible.
reviewed by: Anton Deitmar
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 4132268
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: discrete subgroup
zbMATH Keywords: generalized Poincaré series
zbMATH Keywords: weak Whittaker functions
zbMATH Keywords: automorphic forms
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: functional equation
zbMATH Keywords: higher rank
zbMATH Keywords: Koecher principle
author: R. J. Miatello
author: Nolan Wallach
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90059-1
OpenAlex ID: W1984739925
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description: scientific article; zbMATH DE number 4132268